Characterization of the forcing and sub-filter scale terms in the volume-filtering immersed boundary method

TL;DR

The paper extends the Volume-Filtered Immersed Boundary (VF-IB) method by deriving and characterizing the boundary-forcing term $F_I$ and the sub-filter scale term $\tau_{\mathrm{sfs}}$ that arise during volume-filtering of a hyperbolic PDE with an interface. By analyzing a 2D varying-coefficient hyperbolic equation with a circular interface, the authors show that $F_I$ scales as $1/\delta_f$ while $\tau_{\mathrm{sfs}}$ scales as $\delta_f^2$, leading to second-order convergence with decreasing filter width. Through apriori and aposteriori analyses, they demonstrate that finer interfaces (smaller $\delta_f$) yield results close to the filtered analytic solution and that $\tau_{\mathrm{sfs}}$ can be neglected for sufficiently sharp interfaces, while coarse filters benefit from modeling $\tau_{\mathrm{sfs}}$. The work provides a robust framework for applying VF-IB to topologically complex interfaces and potentially reduces computational cost by enabling accurate simulations on coarser grids when $\tau_{\mathrm{sfs}}$ is appropriately modeled or negligible.

Abstract

We present a characterization of the forcing and the sub-filter scale terms produced in the volume-filtering immersed boundary (VF-IB) method by Dave et al, JCP, 2023. The process of volume-filtering produces bodyforces in the form of surface integrals to describe the boundary conditions at the interface. Furthermore, the approach also produces unclosed subfilter scale (SFS) terms. The level of contribution from SFS terms on the numerical solution depends on the filter width. In order to understand these terms better, we take a 2 dimensional, varying coefficient hyperbolic equation shown by Brady & Liverscu, JCP, 2021. This case is chosen for two reasons. First, the case involves 2 distinct regions seperated by an interface, making it an ideal case for the VF-IB method. Second, an existing analytical solution allows us to properly investigate the contribution from SFS term for varying filter sizes. The latter controls how well resolved the interface is. The smaller the filter size, the more resolved the interface will be. A thorough numerical analysis of the method is presented, as well as the effect of the SFS term on the numerical solution. In order to perform a direct comparison, the numerical solution is compared to the filtered analytical solution. Through this, we highlight three important points. First, we present a methodical approach to volume filtering a hyperbolic PDE. Second, we show that the VF-IB method exhibits second order convergence with respect to decreasing filter size (i.e. making the interface sharper). Finally, we show that the SFS term scales with square the filter size. Large filter sizes require modeling the SFS term. However, for sufficiently finer filters, the SFS term can be ignored without any significant reduction in the accuracy of solution.

Figures (22)

• Figure 1: Computational domain for the test case of a varying coefficient hyperbolic equation similar to the domain used by bradyFoundationsHighorderConservative2021.
• Figure 2: Analytical solution for a varying coefficient hyperbolic equation. The isocontours of $u$ at different time periods $T$ (left) and the value of $u$ vs $x$ in the horizontal direction, centered in the vertical direction (right). There is no velocity field within the region inside the circle.
• Figure 3: Illustration of the volume-filtering approach. Filtering the point-wise fields allows the extraction of the volume fraction $\alpha$, and the averaged point-wise fields $(\alpha\overline{u})$ for region 1. The averaging is performed within a region of size $\delta_f$. The immersed boundary is well resolved when the characteristic corrugation scale $\delta_c$ of the interface is much larger than the filter width $\delta_f$. ($\delta_f$ is not to scale in the figure, but rather shown much larger for easier understanding of the concept of volume-filtering).
• Figure 4: Volume fraction $\alpha$, at three different filter widths with respect to the circle diameter ($\delta_f/D = 1$, $1/2$ and $1/6$). The black contour line represents the Immersed Boundary (IB) surface located at $\alpha = 0.5$ in the limit $\delta_f/D \rightarrow 0$.
• Figure 5: Isocontours of the filtered solution $(\alpha\overline{u})_e$, and the unfiltered solution $u_\mathrm{e}$ at time period $T = 1/4$. At this time the forcing is at its absolute maximum value. $(\alpha\overline{u})_e$ is shown at 2 different filter resolution of $\delta_f/D = 1$ and $\delta_f/D = 1/9$.